nLab
semiprojective morphism

Semiprojective morphisms

Definitions

Let A,BA,B be separable C*C^\ast-algebras. A morphism f:A→Bf\colon A \to B is semiprojective if for any separable C*C^\ast-algebra CC, any increasing sequence Jn⊂CJ_n \subset C of ideals with J=∪n=0∞Jn¯J = \overline{\cup_{n=0}^\infty J_n} and any morphism σ:B→C/J\sigma\colon B \to C/J, there exist nn and an “ff-relative lift” σ˜:A→C/Jn\tilde{\sigma}\colon A \to C/J_n in the sense that the composition A→σ˜C/Jn→C/JA \stackrel{\tilde\sigma} \to C/J_n \to C/J equals the composition A→fB→σC/JA \stackrel{f}\to B \stackrel{\sigma}\to C/J, where C/Jn→C/JC/J_n\to C/J is the epimorphism induced by the inclusion Jn⊂JJ_n \subset J of ideals.

A separable C*C^\ast-algebra is semiprojective if the identity idA:A→Aid_A\colon A \to A is a semiprojective morphism. In particular, every projective separable C*C^\ast-algebra is semiprojective. They are viewed as a generalization of (continuous function algebras) of ANR?s for metric spaces.

This notion is used in the strong shape theory for separable C*C^\ast-algebras.